108156.pdf

Solutions 277

3.11Since the quarterly returnsK(1),K(2),K(3),K(4) are independent and iden-

tically distributed,

E(K(1)) =E(K(2)) =E(K(3)) =E(K(4))

and

1+E(K(0,3)) = (1 +E(K(1))^3 ,

1+E(K(0,4)) = (1 +E(K(1))^4.

Thus, ifE(K(0,3)) = 12%, then the expected quarterly returnE(K(1))∼=

3 .85% and the expected annual returnE(K(0,4))∼= 16 .31%.

3.12By Condition 3.1 the random variables

S(1)

S(0)=1+K(1),

S(2)

S(1)=1+K(2),

S(3)

S(2)=1+K(3) (S.4)

are independent, each taking two values 1 +dand 1 +uwith probabilitiesp

and 1−p, respectively.

The priceS(2), which is the product ofS(0) and the first two of these

random variables, takes up to four values corresponding to the four price

movement scenarios, that is, paths through the two-step tree of stock prices

shown in Figure 3.3 (in whichS(0) = 1 for simplicity). Among these four

values ofS(2) there are in fact only three different ones,

S(2) =







S(0)(1 +u)^2 with probabilityp^2 ,

S(0)(1 +u)(1 +d) with probability 2p(1−p),

S(0)(1 +d)^2 with probability (1−p)^2.

The priceS(3), which is the product ofS(0) and the three independent

random variables in (S.4), takes up to eight values corresponding to the eight

price movement scenarios, that is, paths through the three-step tree of stock

prices in Figure 3.4 (withS(0) = 1 for simplicity). Among these eight values

ofS(3) there are only four different ones,

S(3) =







S(0)(1 +u)^3 with probabilityp^3 ,

S(0)(1 +u)^2 (1 +d) with probability 3p^2 (1−p),

S(0)(1 +u)(1 +d)^2 with probability 3p(1−p)^2 ,

S(0)(1 +d)^3 with probability (1−p)^3.

3.13The top values ofS(1) andS(2) can be used to findu:

u=^9287 −^87 ∼= 0 .0575.

Next,uand the top value ofS(1) give the value ofS(0):

S(0)∼=1+0^87. 0575 ∼= 82. 27

dollars. Finally,dis determined byS(0) and the bottom value ofS(1):

d∼=^7682 −.^8227.^27 ∼=− 0. 0762.